ECMT3150: The Econometrics of Financial Markets (Semester 1, 2023) Tutorial 1b
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ECMT3150: The Econometrics of Financial Markets
(Semester 1, 2023)
Tutorial 1b
1. [R corner] Continuing with the simulation exercise in tutorial 1a, let us now study the ARMA(1,1) model yt = oyt — 1 + ut + θut — 1 , where ut are iid N(0; 1), o = 0:9 and θ = 0:9.
(a) Simulate Öve sample trajectories, each of length 100, from the ARMA(1,1) model. Plot the sample trajectories.
(b) Plot the sample ACF and PACF based on the Örst sample trajectory. Do they make sense based on your understanding on the ARMA(1,1) model?
2. [Mid-sem exam 2019s1] Consider the time series model yt = 0:5 + "t + 0:8"t —2 , where "t ~ wn(0;a = 0:16).
(a) (2 marks) Is the time series stationary? Explain.
(b) (2 marks) Is the time series invertible? Explain.
(c) (1 mark) What is the unconditional mean of yt ?
(d) (2 marks) Compute the unconditional variance of yt .
(e) (4 marks) Derive the autocorrelation function (ACF) of yt . Plot the ACF as a function
of lag order.
(f) (4 marks) Suppose you are now at time t = 100.
i. Compute the 10-step ahead point forecast y^100 (10).
ii. Compute the standard deviation of the forecast error for the point forecast y^100 (10) in (i).
3. Recall the concepts of stationarity and invertibility as discussed in the lecture.
(a) What is the invertibility condition of an MA(2) model?
(b) Is a stationary AR(p) invertible?
(c) Is an invertible MA(q) stationary?
4. Consider the MA(q) model:
yt = 9 + "t + θ1 "t — 1 + ... + θq "t —q ;
where {"t } ~ wn(0;a). Show that the ACF is given by, for j > 0,
pj = Corr(yt ;yt —j ) = { 9ó 1(+)
1(9)9(ó)
and p —j = pj .
5. [Assignment 1 2020s2] For any time series {ut }, deÖne Aut := ut _ ut — 1 and A2 ut := Aut _ Aut — 1 = ut _ 2ut — 1 + ut —2 .
(a) Suppose xt is an MA(1) process given by xt = "t + θ"t — 1 , where lθl < 1 and "t ~ wn(0;a). Show that Axt is an MA(2) process.
(b) Suppose yt is a random walk process deÖned by yt = yt — 1 + "t , where "t ~ wn(0;a). What process does Ayt follow? How about A2 yt ? Show your steps.
(c) Suppose zt is an ARMA(1,1) process given by zt = ozt — 1 + "t + θ"t — 1 , where lol < 1, lθl < 1, and "t ~ wn(0;a). What process does Azt follow? How about A2 zt ? Show your steps.
6. [Assignment 1 2020s2] (Ex: 2.13 of RT) Consider the monthly log returns of CRSP equal- weighted index from January 1962 to December 1999 for 456 observations. You may obtain the data from the Öle m-ew6299 .txt.
(a) Build an AR model for the series and check the Ötted model.
(b) Build an MA model for the series and check the Ötted model.
(c) Compute 1- and 2-step-ahead forecasts of the AR and MA models built in the previous two sub-questions.
(d) Compare the Ötted AR and MA models.
7. [Assignment 1 2021s2] (Ex: 2.8 of RT) Consider the data set of the previous question (i.e., Ex: 2.7 of RT, with data set d-ibm3dxwkdays8008 .txt), but focus on the daily simple returns of the S&P composite index.1 For convenient reporting of parameters estimates, please express the daily simple returns in percentage (i.e., multiply by 100). Perform the necessary data analysis and statistical tests using the 5% signiÖcance level to answer the following questions:
(a) Is there any weekday e§ect on the daily simple returns of the S&P composite index? You may employ a linear regression model to answer this question. Estimate the model, check its validity, and test the hypothesis that there is no Friday e§ect. Draw your conclusion.
[Hint: note that the Öve weekday dummies are collinear. You may either drop one of them, or include all of them in a regression without intercept (implemented using lm(y~x1+x2+ . . .+xn+0)). The latter approach is more convenient for testing the Friday e§ect.]
(b) Check the residual serial correlations using Q(12) statistic. Are there any signiÖcant serial correlations in the residuals? If yes, build a regression model with time series errors for the data.
[Hint: If you use auto .arima, the option xreg=c(var1,var2, . . .) allows you to include regressor variables. If you want to include all Öve weekday dummies, you need to suppress the intercept using the option allowmean=F to avoid collinearity.]
2023-04-05